Theorem lspsolv 21082

Description: If 𝑋 is in the span of 𝐴 ∪ {𝑌} but not 𝐴, then 𝑌 is in the span of 𝐴 ∪ {𝑋}. (Contributed by Mario Carneiro, 25-Jun-2014.)

Hypotheses

Ref	Expression
lspsolv.v	⊢ 𝑉 = (Base‘𝑊)
lspsolv.s	⊢ 𝑆 = (LSubSp‘𝑊)
lspsolv.n	⊢ 𝑁 = (LSpan‘𝑊)

Assertion

Ref	Expression
lspsolv	⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑌 ∈ (𝑁‘(𝐴 ∪ {𝑋})))

Proof of Theorem lspsolv

Dummy variables 𝑟 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lspsolv.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
2		lspsolv.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝑊)
3		lspsolv.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑊)
4		eqid 2733	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2733	. . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2733	. . 3 ⊢ (+g‘𝑊) = (+g‘𝑊)
7		eqid 2733	. . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
8		eqid 2733	. . 3 ⊢ {𝑧 ∈ 𝑉 ∣ ∃𝑟 ∈ (Base‘(Scalar‘𝑊))(𝑧(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴)} = {𝑧 ∈ 𝑉 ∣ ∃𝑟 ∈ (Base‘(Scalar‘𝑊))(𝑧(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴)}
9		lveclmod 21042	. . . 4 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
10	9	adantr 480	. . 3 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑊 ∈ LMod)
11		simpr1 1195	. . 3 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝐴 ⊆ 𝑉)
12		simpr2 1196	. . 3 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑌 ∈ 𝑉)
13		simpr3 1197	. . . 4 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))
14	13	eldifad 3910	. . 3 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑋 ∈ (𝑁‘(𝐴 ∪ {𝑌})))
15	1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14	lspsolvlem 21081	. 2 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → ∃𝑟 ∈ (Base‘(Scalar‘𝑊))(𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))
16	4	lvecdrng 21041	. . . . . . 7 ⊢ (𝑊 ∈ LVec → (Scalar‘𝑊) ∈ DivRing)
17	16	ad2antrr 726	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (Scalar‘𝑊) ∈ DivRing)
18		simprl 770	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑟 ∈ (Base‘(Scalar‘𝑊)))
19	10	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑊 ∈ LMod)
20	12	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑌 ∈ 𝑉)
21		eqid 2733	. . . . . . . . . . . . 13 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
22		eqid 2733	. . . . . . . . . . . . 13 ⊢ (0g‘𝑊) = (0g‘𝑊)
23	1, 4, 7, 21, 22	lmod0vs 20830	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = (0g‘𝑊))
24	19, 20, 23	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = (0g‘𝑊))
25	24	oveq2d 7368	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) = (𝑋(+g‘𝑊)(0g‘𝑊)))
26	11	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝐴 ⊆ 𝑉)
27	20	snssd 4760	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → {𝑌} ⊆ 𝑉)
28	26, 27	unssd 4141	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝐴 ∪ {𝑌}) ⊆ 𝑉)
29	1, 3	lspssv 20918	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ {𝑌}) ⊆ 𝑉) → (𝑁‘(𝐴 ∪ {𝑌})) ⊆ 𝑉)
30	19, 28, 29	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑁‘(𝐴 ∪ {𝑌})) ⊆ 𝑉)
31	30	ssdifssd 4096	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)) ⊆ 𝑉)
32	13	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))
33	31, 32	sseldd 3931	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑋 ∈ 𝑉)
34	1, 6, 22	lmod0vrid 20828	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑋(+g‘𝑊)(0g‘𝑊)) = 𝑋)
35	19, 33, 34	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)(0g‘𝑊)) = 𝑋)
36	25, 35	eqtrd 2768	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) = 𝑋)
37	36, 32	eqeltrd 2833	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))
38	37	eldifbd 3911	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ¬ (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))
39		simprr 772	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))
40		oveq1 7359	. . . . . . . . . . 11 ⊢ (𝑟 = (0g‘(Scalar‘𝑊)) → (𝑟( ·𝑠 ‘𝑊)𝑌) = ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌))
41	40	oveq2d 7368	. . . . . . . . . 10 ⊢ (𝑟 = (0g‘(Scalar‘𝑊)) → (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) = (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)))
42	41	eleq1d 2818	. . . . . . . . 9 ⊢ (𝑟 = (0g‘(Scalar‘𝑊)) → ((𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴) ↔ (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴)))
43	39, 42	syl5ibcom 245	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑟 = (0g‘(Scalar‘𝑊)) → (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴)))
44	43	necon3bd 2943	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (¬ (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴) → 𝑟 ≠ (0g‘(Scalar‘𝑊))))
45	38, 44	mpd 15	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑟 ≠ (0g‘(Scalar‘𝑊)))
46		eqid 2733	. . . . . . 7 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
47		eqid 2733	. . . . . . 7 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
48		eqid 2733	. . . . . . 7 ⊢ (invr‘(Scalar‘𝑊)) = (invr‘(Scalar‘𝑊))
49	5, 21, 46, 47, 48	drnginvrl 20673	. . . . . 6 ⊢ (((Scalar‘𝑊) ∈ DivRing ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑟 ≠ (0g‘(Scalar‘𝑊))) → (((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟) = (1r‘(Scalar‘𝑊)))
50	17, 18, 45, 49	syl3anc 1373	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟) = (1r‘(Scalar‘𝑊)))
51	50	oveq1d 7367	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟)( ·𝑠 ‘𝑊)𝑌) = ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌))
52	5, 21, 48	drnginvrcl 20670	. . . . . 6 ⊢ (((Scalar‘𝑊) ∈ DivRing ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑟 ≠ (0g‘(Scalar‘𝑊))) → ((invr‘(Scalar‘𝑊))‘𝑟) ∈ (Base‘(Scalar‘𝑊)))
53	17, 18, 45, 52	syl3anc 1373	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((invr‘(Scalar‘𝑊))‘𝑟) ∈ (Base‘(Scalar‘𝑊)))
54	1, 4, 7, 5, 46	lmodvsass 20822	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (((invr‘(Scalar‘𝑊))‘𝑟) ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ 𝑉)) → ((((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟)( ·𝑠 ‘𝑊)𝑌) = (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)))
55	19, 53, 18, 20, 54	syl13anc 1374	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟)( ·𝑠 ‘𝑊)𝑌) = (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)))
56	1, 4, 7, 47	lmodvs1 20825	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = 𝑌)
57	19, 20, 56	syl2anc 584	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = 𝑌)
58	51, 55, 57	3eqtr3d 2776	. . 3 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) = 𝑌)
59	33	snssd 4760	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → {𝑋} ⊆ 𝑉)
60	26, 59	unssd 4141	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝐴 ∪ {𝑋}) ⊆ 𝑉)
61	1, 2, 3	lspcl 20911	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ {𝑋}) ⊆ 𝑉) → (𝑁‘(𝐴 ∪ {𝑋})) ∈ 𝑆)
62	19, 60, 61	syl2anc 584	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑁‘(𝐴 ∪ {𝑋})) ∈ 𝑆)
63	1, 4, 7, 5	lmodvscl 20813	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ 𝑉) → (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉)
64	19, 18, 20, 63	syl3anc 1373	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉)
65		eqid 2733	. . . . . . 7 ⊢ (-g‘𝑊) = (-g‘𝑊)
66	1, 6, 65	lmodvpncan 20850	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋)(-g‘𝑊)𝑋) = (𝑟( ·𝑠 ‘𝑊)𝑌))
67	19, 64, 33, 66	syl3anc 1373	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋)(-g‘𝑊)𝑋) = (𝑟( ·𝑠 ‘𝑊)𝑌))
68	1, 6	lmodcom 20843	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋) = (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)))
69	19, 64, 33, 68	syl3anc 1373	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋) = (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)))
70		ssun1 4127	. . . . . . . . . 10 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑋})
71	70	a1i 11	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝐴 ⊆ (𝐴 ∪ {𝑋}))
72	1, 3	lspss 20919	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ {𝑋}) ⊆ 𝑉 ∧ 𝐴 ⊆ (𝐴 ∪ {𝑋})) → (𝑁‘𝐴) ⊆ (𝑁‘(𝐴 ∪ {𝑋})))
73	19, 60, 71, 72	syl3anc 1373	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑁‘𝐴) ⊆ (𝑁‘(𝐴 ∪ {𝑋})))
74	73, 39	sseldd 3931	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
75	69, 74	eqeltrd 2833	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
76	1, 3	lspssid 20920	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ {𝑋}) ⊆ 𝑉) → (𝐴 ∪ {𝑋}) ⊆ (𝑁‘(𝐴 ∪ {𝑋})))
77	19, 60, 76	syl2anc 584	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝐴 ∪ {𝑋}) ⊆ (𝑁‘(𝐴 ∪ {𝑋})))
78		snidg 4612	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
79		elun2 4132	. . . . . . . 8 ⊢ (𝑋 ∈ {𝑋} → 𝑋 ∈ (𝐴 ∪ {𝑋}))
80	33, 78, 79	3syl 18	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑋 ∈ (𝐴 ∪ {𝑋}))
81	77, 80	sseldd 3931	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑋 ∈ (𝑁‘(𝐴 ∪ {𝑋})))
82	65, 2	lssvsubcl 20879	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ (𝑁‘(𝐴 ∪ {𝑋})) ∈ 𝑆) ∧ (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋) ∈ (𝑁‘(𝐴 ∪ {𝑋})) ∧ 𝑋 ∈ (𝑁‘(𝐴 ∪ {𝑋})))) → (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋)(-g‘𝑊)𝑋) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
83	19, 62, 75, 81, 82	syl22anc 838	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋)(-g‘𝑊)𝑋) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
84	67, 83	eqeltrrd 2834	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
85	4, 7, 5, 2	lssvscl 20890	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ (𝑁‘(𝐴 ∪ {𝑋})) ∈ 𝑆) ∧ (((invr‘(Scalar‘𝑊))‘𝑟) ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ (𝑁‘(𝐴 ∪ {𝑋})))) → (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
86	19, 62, 53, 84, 85	syl22anc 838	. . 3 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
87	58, 86	eqeltrrd 2834	. 2 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑌 ∈ (𝑁‘(𝐴 ∪ {𝑋})))
88	15, 87	rexlimddv 3140	1 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑌 ∈ (𝑁‘(𝐴 ∪ {𝑋})))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  ∃wrex 3057  {crab 3396   ∖ cdif 3895   ∪ cun 3896   ⊆ wss 3898  {csn 4575  ‘cfv 6486  (class class class)co 7352  Basecbs 17122  +_gcplusg 17163  ._rcmulr 17164  Scalarcsca 17166   ·_𝑠 cvsca 17167  0_gc0g 17345  -_gcsg 18850  1_rcur 20101  inv_rcinvr 20307  DivRingcdr 20646  LModclmod 20795  LSubSpclss 20866  LSpanclspn 20906  LVecclvec 21038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-tpos 8162  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-2 12195  df-3 12196  df-sets 17077  df-slot 17095  df-ndx 17107  df-base 17123  df-ress 17144  df-plusg 17176  df-mulr 17177  df-0g 17347  df-mgm 18550  df-sgrp 18629  df-mnd 18645  df-grp 18851  df-minusg 18852  df-sbg 18853  df-cmn 19696  df-abl 19697  df-mgp 20061  df-rng 20073  df-ur 20102  df-ring 20155  df-oppr 20257  df-dvdsr 20277  df-unit 20278  df-invr 20308  df-drng 20648  df-lmod 20797  df-lss 20867  df-lsp 20907  df-lvec 21039

This theorem is referenced by:  lssacsex  21083  lspsnat  21084  lsppratlem1  21086  lsppratlem3  21088  lsppratlem4  21089  lbsextlem4  21100  lindsadd  37673  lindsenlbs  37675